a. Matrix A is invertible when |A| = -m ≠ 0 then statement true.
b. The system AX = B has no solution when m = 0 and n ≠ 0 has a real number then statement true.
c. The system AX = B has an infinite number of solutions when m = n = 0 then statement true.
d. Columns of the augmented matrix (AB) are linearly independent when m ≠ 0 and n= 0 then statement true.
e. The system AX = 0 has a unique solution when m ≠ 0 then statement true.
f. At least one eigenvalue of the matrix A is zero when m = 0 then statement true.
g. Columns of the matrix A form a basis in R³ when m ≠ 0 then statement true.
Given that,
The augmented matrix of a system of linear equations AX = B was reduced to upper-triangular form so that
[A|B] = [tex]\left[\begin{array}{ccc}2&1&0 \ | \ 2\\0&-1&3 \ | \ 1 \\0&0&m \ | \ n\end{array}\right][/tex]
Where m and n are real numbers.
We know that,
a. We have to prove matrix A is invertible.
For A to be invertible.
|A| ≠ 0
|A| is the determinant of the matrix A.
|A| = 2(-m) -1(0) + 0(0) = -m
Here, m is the real number.
So, |A| = -m ≠ 0
Therefore, Matrix A is invertible when |A| = -m ≠ 0 then statement true.
b. We have to prove the system AX = B has no solution.
When Rank[A|B] > Rank[A]
m = 0 and n ≠ 0 has a real number
Therefore, The system AX = B has no solution when m = 0 and n ≠ 0 has a real number then statement true.
c. We have to prove the system AX = B has an infinite number of solutions.
When m = n = 0, and Rank[A] < 3
Therefore, The system AX = B has an infinite number of solutions when m = n = 0 then statement true.
d. We have to prove columns of the augmented matrix (AB) are linearly independent.
When m ≠ 0 and m∈R and n= 0
Therefore, Columns of the augmented matrix (AB) are linearly independent when m ≠ 0 and n= 0 then statement true.
e. We have to prove the system AX = 0 has a unique solution.
When [tex]\left[\begin{array}{ccc}2&1&0 \\0&-1&3 \\0&0&m \end{array}\right]\left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}0\\0\\0\end{array}\right][/tex]
The equation are 2x + y = 0, -y + 3z = 0 and mz = 0
m ≠ 0 should be any real number except zero.
Therefore, The system AX = 0 has a unique solution when m ≠ 0 then statement true.
f. We have to prove at least one eigenvalue of the matrix A is zero.
When λ = 2, 1, m
m = 0 then eigen value is zero
Therefore, At least one eigenvalue of the matrix A is zero when m = 0 then statement true.
g. We have to prove columns of the matrix A form a basis in R³.
When m ≠ 0
Therefore, Columns of the matrix A form a basis in R³ when m ≠ 0 then statement true.
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How to find volume of a cylinder by using the same dimensions of a cone but only knowing the volume of the cone
Answer:
Multiply the volume of the cone by 3
Step-by-step explanation:
Required
Volume of a cylinder from a cone of the same dimension
The volume of a cone is:
[tex]V_1 = \frac{1}{3}\pi r^2h[/tex]
The volume of a cylinder is:
[tex]V_2 = \pi r^2h[/tex]
Recall that:
[tex]V_1 = \frac{1}{3}\pi r^2h[/tex]
Split:
[tex]V_1 = \frac{1}{3}*[\pi r^2h][/tex]
Substitute: [tex]V_2 = \pi r^2h[/tex]
[tex]V_1 = \frac{1}{3}*V_2[/tex]
Make V2 the subject
[tex]V_2 =3V_1[/tex]
This implies that, we simply multiply the volume of the cone by 3
write an expression describing all the angles that are coterminal with 358°. (please use the variable in your answer. give your answer in degrees, but do not include a degree symbol in your answer.)
Answer: 358 + 360n where n is an integer
Reason:
Coterminal angles point in the same direction.
We add on multiples of 360 to rotate a full circle, and we get back to the same direction that 358 degrees points in (almost directly to the east). The variable n is an integer {..., -3, -2, -1, 0, 1, 2, 3, ...}
If n is negative, then we subtract off multiples of 360.
If f(x) = 2x³ + Ax² +8x-3 and f(2)= 1, what is the value of A?
the value of A in the function f(x) = 2x³ + Ax² + 8x - 3 is -7.
To find the value of A in the function f(x) = 2x³ + Ax² + 8x - 3, we are given that f(2) = 1. Substituting x = 2 into the function, we have:
f(2) = 2(2)³ + A(2)² + 8(2) - 3
Simplifying further:
1 = 2(8) + 4A + 16 - 3
1 = 16 + 4A + 13
Combining like terms:
1 = 29 + 4A
To isolate A, we subtract 29 from both sides:
-28 = 4A
Finally, we divide both sides by 4 to solve for A:
A = -7
Therefore, the value of A in the function f(x) = 2x³ + Ax² + 8x - 3 is -7.
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Using the greatest common factor (GCF), what is the factored form of: 24v-36
Answer:
12(2v - 3)
Step-by-step explanation:
GCF of 24 and 36 is 12
24v - 36
12(2v - 3)
word form? 132 = 132
Answer:
one hundred thirty-two
Step-by-step explanation:
should be it if it is can i get brainliest?
Which of the following is not a solution for finding the perimeter of the square?
Answer:
y
Step-by-step explanation:
y needs to provide a number
What are the next three terms of this pattern?
- 4, - 7, - 2, -5, 0, __, __, __
Answer:
-4, -7, -2, -5, 0, -3, 2, -1
Step-by-step explanation:
the pattern is -3, +5.
i hope this helps :)
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = 0 x < 0 x2 16 0 ? x ? 4 1 4 ? x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ? 1). (b) Calculate P(0.5 ? X ? 1). (c) Calculate P(X > 1.5). (d) What is the median checkout duration mu tilde? [solve 0.5 = F(mu tilde)]. (e) Obtain the density function f(x). f(x) = F?'(x) = (f) Calculate E(X). (g) Calculate V(X) and ?x. V(X) = ?x = (h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].
X denote the amount of time a book on two-hour reserve is actually checked out, and for the given cdf the following are the answers for the questions asked
(a) P(X ≤ 1) ≈ 0.0625
(b) P(0.5 ≤ X ≤ 1) ≈ 0.0469
(c) P(X > 1.5) ≈ 0.8594
(d) Median checkout duration ≈ 2.828
(e) Density function f(x) is defined as:
f(x) = 0 for x < 0
f(x) = x/8 for 0 ≤ x ≤ 4
f(x) = 0 for x > 4
(f) Expected value E(X) ≈ 2.667
(g) Variance V(X) ≈ 0.889, Standard Deviation σ(X) ≈ 0.943
(h) Expected charge E[h(X)] = 8
In probability theory, a cumulative distribution function (CDF) provides information about the probabilities of certain events occurring in a random variable. In this scenario, let's consider a book that is on a two-hour reserve and denote the amount of time it is checked out as X. We are given the CDF of X and we will use it to calculate various probabilities and statistics related to the checkout duration of the book.
The given CDF is as follows:
F(x) = 0 for x < 0
F(x) = x²/16 for 0 ≤ x ≤ 4
F(x) = 1 for x > 4
(a) P(X ≤ 1):
To calculate this probability, we need to find F(1) since F(x) represents the cumulative probability up to x. From the given CDF, we see that F(x) = x²/16 for 0 ≤ x ≤ 4. Substituting x = 1 into the equation, we get:
F(1) = (1²)/16 = 1/16.
(b) P(0.5 ≤ X ≤ 1):
To calculate this probability, we need to find F(1) - F(0.5) since F(x) represents the cumulative probability up to x. From the given CDF, we have F(0.5) = (0.5²)/16 = 1/64 and F(1) = (1²)/16 = 1/16. Therefore,
P(0.5 ≤ X ≤ 1) = F(1) - F(0.5) = (1/16) - (1/64) = 3/64.
(c) P(X > 1.5):
To calculate this probability, we need to find 1 - F(1.5) since F(x) represents the cumulative probability up to x. From the given CDF, we have F(1.5) = (1.5²)/16 = 9/64. Therefore,
P(X > 1.5) = 1 - F(1.5) = 1 - (9/64) = 55/64.
(d) Median checkout duration:
The median is the value that divides the distribution into two equal parts, meaning that half of the checkouts are below this value and half are above it. We need to solve the equation F(median) = 0.5. From the given CDF, we have:
F(median) = 0.5
0.5 = (median²)/16
Solving for the median, we get median = √(8) ≈ 2.828.
(e) Density function f(x):
The density function f(x) represents the derivative of the cumulative distribution function F(x). To obtain f(x), we differentiate the given CDF:
f(x) = F'(x)
For x < 0, f(x) = 0 since F(x) is constant in that range.
For 0 ≤ x ≤ 4, we have F(x) = x²/16.
Differentiating with respect to x, we get:
f(x) = d/dx (x²/16) = (2x)/16 = x/8.
For x > 4, f(x) = 0 since F(x) is constant in that range.
Therefore, the density function f(x) is:
f(x) = 0 for x < 0
f(x) = x/8 for 0 ≤ x ≤ 4
f(x) = 0 for x > 4
(f) Expected value E(X):
The expected value of a random variable X is a measure of its average value. To calculate E(X), we integrate the product of x and the density function f(x) over the entire range of X:
E(X) = ∫[x * f(x)] dx
For x < 0 and x > 4, f(x) = 0, so we only need to consider the interval 0 ≤ x ≤ 4:
E(X) = ∫[x * (x/8)] dx
= (1/8) ∫[x²] dx (integrating x²)
= (1/8) * (x³/3) + C (integrating x²)
= (1/24) * (x³) + C
Evaluating this expression from x = 0 to x = 4, we get:
E(X) = (1/24) * (4³) - (1/24) * (0³)
= 64/24
= 8/3
≈ 2.667
(g) Variance V(X) and Standard Deviation σ(X):
Variance is a measure of the spread or dispersion of a random variable. To calculate V(X), we need to calculate the second moment E(X²) and subtract the square of the expected value [E(X)]². The standard deviation σ(X) is the square root of the variance.
E(X²):
E(X²) = ∫[x² * f(x)] dx
For x < 0 and x > 4, f(x) = 0, so we only need to consider the interval 0 ≤ x ≤ 4:
E(X²) = ∫[x² * (x/8)] dx
= (1/8) ∫[x³] dx (integrating x³)
= (1/8) * (x⁴/4) + C (integrating x³)
= (1/32) * (x^4) + C
Evaluating this expression from x = 0 to x = 4, we get:
E(X²) = (1/32) * (4⁴) - (1/32) * (0⁴)
= 256/32
= 8
V(X):
V(X) = E(X²) - [E(X)]²
= 8 - (8/3)²
= 8 - 64/9
= 8 - 7.111
≈ 0.889
Standard deviation:
σ(X) = √(V(X))
= √(0.889)
≈ 0.943
(h) Expected charge E[h(X)]:
Given the function h(X) = X², we want to calculate the expected value of h(X). This can be done by finding E[h(X)] = E(X²).
From the previous calculations, we know that E(X²) = 8. Therefore, the expected charge is E[h(X)] = 8.
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The total cost of a catered meal for the Armstrong family reunion was to be split equally by the 20 people who came. Before the meal started, five more Armstrong's unexpectedly appeared. The total cost was shared equally among the 25 people, and so each person in the original group owed $4 less. What was the total cost of the catered meal?
Answer:
$400
Step-by-step explanation:
Let the initial total cost be represented = $X.
Hence, for 20 family members
= $ x/20 each .......1
Now that they are 25 members
= $x/25 each.......2
Each in the original group which is equation 1 owed $4 less.
That is,
x/20 - 4 ........ 3
Now equate 2 and 3
X/20 - 4/1 = X/25
Taking the LCM of both sides
X-80/20 = X/25
Cross multiply both sides
25( x-80 ) = 20x
25x-2000 = 20x
Collecting like terms
25x-20x = 2000
5x = 2000
Divide through by 5
X = 2000/5
X = 400
The total cost for the catered meal is $400.
To check if it was correct
Put 400 into equation 1 and 2 and then minus whatever you get from each other, you will get $4
From equation 1
400/20 = $20
From equation 2
400/25 =$16
$20-$16 = $4
Thanks
(a) Find the determinant of the given matrix M. Show all your work. 2 2 -1 9 M= 4 2 8 8 1 17 24 0 10-13 1 determinant of the matrix -2B³A-CTB-¹AT.
(b) Let A, B, C be 3 x 3 matrices with det A= -2
In part (a) of the problem, the task is to find the determinant of the given matrix M by showing all the steps of the calculation. In part (b), we are given three matrices A, B, and C with det A = -2.
(a) To find the determinant of matrix M, we can use various methods such as cofactor expansion or row operations. Let's use cofactor expansion along the first row:
det M = 2 * (-1)^(1+1) * det [[2 8 0] [17 24 0] [10 -13 1]]
- 2 * (-1)^(1+2) * det [[4 8 0] [1 24 0] [10 -13 1]]
Simplifying and evaluating the determinants of the 2x2 matrices, we get:
det M = 2 * (2 * 24 - 17 * (-13))
- 2 * (4 * 24 - 1 * (-13))
After performing the calculations, we find the determinant of matrix M.
(b) The given information states that det A = -2. However, no specific task or question is mentioned regarding matrices A, B, and C. Further instructions or requirements are needed to provide a specific answer or analysis based on the given information.
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Which expression is equivalent to 2x + 2
A (2 + x) + 2
B 2(x + 2)
C 2(x + 1)
D 4x
Find the volume of this cylinder.
Round to the nearest tenth.
8ft
-5ft
[?] ft3
Answer:
628 ft
Step-by-step explanation:
The volume of cylinder is, V= 628 ft³.
What is Volume?The amount of space occupied by a three-dimensional figure as measured in cubic units.
Given:
radius,r= 5 ft, height, h = 8 ft
Volume= πr²h
V= 3.14 * 5 *5 * 8
V= 628 ft³
Hence, the volume of cylinder is 628 ft³.
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(PLEASE PLEASE HELP)
Find the area of the triangle.
Answer:
40.2 in²
Step-by-step explanation:
h*b/2
h= 6.7
b= 12
find the volume of the solid enclosed by the surface z = 1 −x2 −y2 and the xy-plane.
The integral becomes:
V = ∫₀¹ ∫₀²π (1 − r²) r dθ dr
To find the volume of the solid enclosed by the surface z = 1 − x² − y² and the xy-plane, we need to integrate the function f(x, y) = 1 − x² − y² over the region in the xy-plane where z is positive.
Since the surface z = 1 − x² − y² represents a downward-opening paraboloid, we can set up the integral as follows:
V = ∬R (1 − x² − y²) dA
Here, R represents the region in the xy-plane bounded by the curve x² + y² = 1. To evaluate this integral, we can switch to polar coordinates. In polar coordinates, the region R corresponds to the interval 0 ≤ θ ≤ 2π and 0 ≤ r ≤ 1. The differential area element dA in polar coordinates is r dr dθ.
Evaluating this double integral will give us the volume of the solid enclosed by the surface and the xy-plane.
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Which of the following numbers is not a
rational number?
A. -3
B. 2.7
C. 14
D. 15
Trick question! All are rational numbers.
..................................................................................
Find a harmonic conjugate v(x, y) of u(x, y) = 2x(1 - y)
The harmonic conjugate of u(x, y) = 2x(1 - y) is v(x, y) = 2y - y² - x² + C, where C is an arbitrary constant.
How to find the harmoic conjugate?Here we want to find the harmonic conjugate of:
u(x, y) = 2x*(1 - y)
To do so, we need to use the Cauchy-Riemann equations state that for a function f(z) = u(x, y) + iv(x, y) to be analytic (holomorphic), the partial derivatives of u and v must satisfy the following conditions:
∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x
Let's find the harmonic conjugate by solving these equations:
Given u(x, y) = 2x(1 - y)
∂u/∂x = 2(1 - y)
∂u/∂y = -2x
Setting these derivatives equal to the respective partial derivatives of v:
∂v/∂y = 2(1 - y)
∂v/∂x = -2x
Now, integrate the first equation with respect to y, treating x as a constant:
v(x, y) = 2y - y² + f(x)
Differentiate the obtained equation with respect to x:
∂v/∂x = f'(x)
Comparing this derivative with the second equation, we have:
f'(x) = -2x
Integrating f'(x) with respect to x:
f(x) = -x² + C
where C is a constant of integration.
Now, substitute f(x) into the equation for v(x, y):
v(x, y) = 2y - y² - x² + C
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- A computer modem can transmit
1.5 X 10 bytes per second. How many
bytes can it transmit in 300 seconds?
Write your answer in scientific notation.
6v=-9 v= what does v =
Answer:
v = -3/2
Step-by-step explanation:
6v = -9
v = -9/6
v = -3/2
I NEED HELP!! with this homework please. For B,C,and D, if you know the answer please tell me I’m kinda confused.
I’ll make you a brainlist for the reward:)
Answer:
Step-by-step explanation:
Location Reflect over x Reflect over y
B -0.75 ,1 -0.75 ,-1 +0.75 ,1
C 1.75 ,-1 1.75,1 -1.75,+1
D -2,-2 -2,+2 +2 ,-2
Which angle is congruent to <4 <1<2<5<8
Answer:
The angle that will be congruent to angle 4 is :
Angle 1
Step-by-step explanation:
It is given that:
angle 4 and angle 5 are complements.
Also, angle 1 and angle 5 are complements.
Congruent complementary Theorem--
It states that if two angles are complementary to the same angle, then the two angles are congruent to each other.
Here both angle 1 and angle 4 are complementary to the same angle i.e. angle 5.
HELPPPPPP!!!!!!!!!!!!!!!!
Answer: [tex]w\geq 20[/tex]
Step-by-step explanation:
Solve this like it's an equation
6w+30>150 (i know it is larger than or equal to)
6w>120
w>120/6
w>20
Answer:
The answer is 20
Step-by-step explanation:
6w+30≥150
6w≥150-30
6w≥120
divide both sides by 6
6w/6≥120/6
w≥20
we can say
w=20 since w is greater than or equal to
What is an appropriate horizontal scale and vertical scale for the viewing window?
a. Horizontal scale: 1 unit = 10, Vertical scale: 1 unit = 5
b. Horizontal scale: 1 unit = 5, Vertical scale: 1 unit = 10
c. Horizontal scale: 1 unit = 1, Vertical scale: 1 unit = 1
d. Horizontal scale: 1 unit = 10, Vertical scale: 1 unit = 10
option D: Horizontal scale: 1 unit = 10, Vertical scale: 1 unit = 10 is the correct answer.
To determine an appropriate horizontal scale and vertical scale for the viewing window, one must analyze the problem, the graph, and the data presented in the problem. The appropriate horizontal scale and vertical scale for the viewing window is given by option D, that is Horizontal scale: 1 unit = 10, Vertical scale: 1 unit = 10. The horizontal scale and vertical scale refer to the scale of the x-axis and y-axis of the graph. These scales are used to represent data in a graphic format that can be easily understood by readers. The horizontal scale is used to determine the value of the x-axis in the graph. The vertical scale is used to determine the value of the y-axis in the graph. The scale should be chosen based on the range of values in the graph. When the values in the graph are too small or too large, the scale must be adjusted accordingly. In this case, since the values of horizontal and vertical scale are large, the scale must be adjusted accordingly. Therefore, option D is the correct answer.
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The size of the horizontal and vertical scales should be chosen so that the graph fits in the viewing window. So, that is why the answer is "b. Horizontal scale: 1 unit = 5, Vertical scale: 1 unit = 10."
When graphing a function, it is important to choose an appropriate horizontal scale and vertical scale for the viewing window. The answer is "b. Horizontal scale: 1 unit = 5, Vertical scale: 1 unit = 10."Why is the answer (b) Horizontal scale: 1 unit = 5, Vertical scale: 1 unit = 10?The answer to this question is based on the graph. It is more appropriate to use (b) horizontal scale: 1 unit = 5, vertical scale: 1 unit = 10. In this situation, the x-axis should have a horizontal scale of 1 unit = 5 so that the x-axis can fit within the viewing window. Meanwhile, the y-axis should have a vertical scale of 1 unit = 10 so that the entire graph can fit in the viewing window.
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WHAT DOES THE WORD MILD MEAN
A HUMID
B MOSTLY COULD
C NO TOO HOT OR TOO COLD
D NEVER THE SAME
Answer:
C NO TOO HOT OR TOO COLD
Step-by-step explanation:
Answer:
the answer is c
Step-by-step explanation:
Brainliest plz
Is it just me or do you guys get annoyed when people give us a whole explanation like, I only need the letter of the answer.
Answer:
Yes lol
Step-by-step explanation:
Well the app says it helps people learn the answers but when I'm in a test i just look at letters:)
Answer:
I mean i'm the type to give a long winded answer. But i guess it depends on the context. Sometimes people ask for an explanation yk? Like most of those ppl who ask questions for History and English need a written response.
If the two equations in a system of linear equations are added and the results is 9x=0 the system has no solution
Answer:
0
Step-by-step explanation:
HOPE I HELPED!
Please help and explain!
Find the total surface area shown down below
Ignore the colors!
HURRY I NEED HELP PLS
Find the volume of each solid. Round to the nearest tenth if necessary.
Answer:
3,454 cm³
Step-by-step explanation:
This is a cylinder so the volume formula is [tex]V=\pi r^{2} h[/tex]
The problem gives us the diameter (20 cm) and the height (11 cm). Since the formula uses the radius instead of the diameter, we have to divide the diameter by 2.
20 ÷ 2 = 10
The radius is 10 cm
Now, we plug our height and radius into the formula to find the volume.
[tex]V=\pi r^{2} h\\V=(\frac{22}{7})(10)^{2}(11)\\V=(\frac{22}{7})(100)(11)\\[/tex]
V≈3,454
in a certain lottery, you must correctly select 5 numbers (in any order) out of 29 to win. you purchase one lottery ticket. what is the probability that you will win?
To calculate the probability of winning a certain lottery where you must select 5 numbers out of 29 in any order, we can use the concept of combinations. The probability of winning can be determined by dividing the number of successful outcomes (winning combinations) by the total number of possible outcomes.
In this lottery, you need to select 5 numbers out of 29, and the order of selection doesn't matter. The total number of possible outcomes is given by the combination formula, which is denoted as C(29, 5) and calculated as
29! / (5! * (29-5)!).
To win, you have only one successful outcome, which is the combination of the 5 winning numbers. Therefore, the probability of winning can be calculated as 1 / C(29, 5). Evaluating this expression will give you the probability of winning the lottery with a single-ticket purchase.
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Karen has $33 to buy pencils for her art school. She buys boxes of pencils that each cost $1.50 and is left with $9 after her purchase. How many boxes of pencils did she buy? Enter your answer in the box below.
Answer:16
Step-by-step explanation:
33-9=24
24/1.50=16
Given that CAT = DOG select all statements that are true,
Answer:
The answer to this problem Is A I think
Step-by-step explanation: